Lefschetz pencil ($f:X⇢{\mathbb{CP}}^1$)

A Lefschetz pencil on an oriented four-manifold is a map $f:X\setminus B\to {\mathbb{CP}}^1$ with a finite base locus $B$, locally modeled by $[z_1:z_2]$ at each base point, and isolated critical points locally modeled by $z_1^2+z_2^2$. Blowing up the base points turns the pencil into a Lefschetz fibration.

On a smooth projective surface, fix an ample divisor $H$. For sufficiently large $m$, a general pencil in $|mH|$ is a Lefschetz pencil. The adjunction formula gives the displayed genus. If $m$ is divisible by four, both terms added to $1$ are even integers, so the genus is odd.

The standard four-torus has trivial complex tangent bundle, so its symplectic canonical class is zero. The symplectic adjunction formula gives $F^2=2g-2$ for a smooth genus-$g$ fiber of a symplectic Lefschetz pencil. Two generic fibers meet once positively at each base point and nowhere else, giving the displayed count.

A Lefschetz fibration is a map to a surface whose critical points have the complex Morse local model $z_1^2+\cdots +z_n^2$. Away from the critical values it is a smooth fiber bundle.

If a Lefschetz pencil on a closed four-manifold $X$ has smooth fiber of genus $g$, $b$ base points and $m$ critical points, thenIndeed, blowing up the base points gives $\chi (\tilde{X})=\chi (X)+b$, while a genus-$g$ fibration over $S^2$ contributes $2(2-2g)$ and each Lefschetz critical point contributes one.

Every symplectic smooth fiber in a Lefschetz pencil on ${\mathbb{CP}}^2$ represents $dH$ for an integer $d\ge 1$. The symplectic adjunction formula givesConversely, generic pencils of degree-$d$ complex plane curves realize every value in this list.